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Full text of "Treatise On Analysis Vol-Ii"


4.    Let E = J~| Ea be a product of Hausdorff topological spaces, such that each E

ae I

contains at least two distinct points aa,ba. For each a e I, let cx be the point of E
such that pra cx = ba and pi> ca == aff for all /3 ^ a. Show that every point of the set
{cjaei is an isolated point.

Deduce that the topology of E has a denumerable basis if and only I is denumer-
able and each of the E has a denumerable basis.

Show that if I is not denumerable, the point a  (<2a) has no denumerable funda-
mental system of neighborhoods. If Ea  {ax, a} for each a, and if F c E is the set of
all x e E such that pra# = bn for all but denumerably many indices a, show that F is
dense in E and that a  (ax) is not a limit of any sequence of elements of F.*

5.    Let K be the discrete space {0,1}, let A be an infinite set, and let E be the product
space KA. Let V be a nonempty elementary set in E. If h is the (finite) number of
indices a such that prV ^ K, let /-t(V) = 2'* (cf. (13.21)).

(a)    Show that, if U1?..., Un are nonempty mutually disjoint elementary sets, then

JT fji(Uk) <; 1. (Write Ufc in the form W* x KB, where B is the same for all k and is the

complement of a finite subset of A.)

(b)    Deduce from (a) that if (UA)A e L is a family of nonempty pairwise disjoint open
sets in E, then L is at most denumerable, although there exist sets C of arbitrary
cardinal in E (for a suitable choice of A) all of whose points are isolated. Show also
that if the cardinal of A is strictly greater than that of $(N), then E contains no
denumerable dense subset.

6.    With the notation of Problem 5, let 93 be the set of all subsets of KA of the form

Y[ Ma, where M = K except for at most denumerably many indices a. Show that 93

ae A

is a basis for a Hausdorff topology on KA which is finer than the product topology,
and which is not discrete provided that A is infinite and non-denumerable. In this topol-
ogy, every denumerable intersection of open sets is open; no point has a denumerable
fundamental system of neighborhoods; and every quasi-compact subset is finite. The
projections pr are continuous with respect to this topology. Deduce that KA is
uniformizable with respect to this topology (Section 12.4, Problem 5).

7.    Let I be the interval [0,1] in R, with the induced topology. Show that every separable
metrizable space E is homeomorphic to a subspace of the product IN. (Reduce to the
case where the distance d on E is <H, and consider a sequence (an) which is dense in
E, and the functions xt-+d(aat x).)

8.    With the notation of Problem 7, show that in the uniformizable product space I1,
the subspace of continuous mappings of I into I is dense. Deduce that I1 has a
denumerable dense subset, although there is no denumerable basis of open sets
(Problem 4).

9.    Show that if I is a nonempty open interval in R, there exists no nonconstant mapping
/of I into the product space NN with the following property: for each x e I and each

* This example shows that in general topological spaces (and even in Hausdorff
uniformizable spaces) the convergent sequences do not determine the topology, as they do
in metrizable spaces (cf. (3.13.3) and (3.13.4)). To get corresponding results in general, it is
necessary to replace the notion of sequence by the more general notion of a filter (cf. [5]).(#) = {x}. For each x G A, let S*,  denote the union of {x} and the set of integers J>/i,